Void regions of the Universe offer a special environment for studying cosmology and galaxy formation, which may expose weaknesses in our understanding of these phenomena. Although galaxies in voids are observed to be predominately gas rich, star forming and blue, a subpopulation of bright-red void galaxies can also be found, whose star formation was shutdown long ago. Are the same processes that quench star formation in denser regions of the Universe also at work in voids? We compare the luminosity function of void galaxies in the 2dF Galaxy Redshift Survey, to those from a galaxy formation model built on the Millennium simulation. We show that a global star formation suppression mechanism in the form of low-luminosity 'radio-mode' active galactic nuclei (AGN) heating is sufficient to reproduce the observed population of void early types. Radio-mode heating is environment independent other than its dependence on dark matter halo mass, where, above a critical mass threshold of approximately Mvir∼ 10(12.5) Mo, gas cooling on to the galaxy is suppressed and star formation subsequently fades. In the Millennium simulation, the void halo mass function is shifted with respect to denser environments, but still maintains a high-mass tail above this critical threshold. In such void haloes, radio-mode heating remains efficient and red galaxies are found; collectively these galaxies match the observed space density without any modification to the model. Consequently, galaxies living in vastly different large-scale environments but hosted by haloes of similar mass are predicted to have similar properties, consistent with observations.